The curve of fastest descent is not a straight or polygonal line (blue) but a cycloid (red).
In physics and mathematics, a brachistochrone curve (from Ancient Greek βράχιστος χρόνος (brákhistos khrónos) 'shortest time'),[1] or curve of fastest descent, is the one lying on the plane between a point A and a lower point B, where B is not directly below A, on which a bead slides frictionlessly under the influence of a uniform gravitational field to a given end point in the shortest time. The problem was posed by Johann Bernoulli in 1696.
The brachistochrone curve is the same shape as the tautochrone curve; both are cycloids. However, the portion of the cycloid used for each of the two varies. More specifically, the brachistochrone can use up to a complete rotation of the cycloid (at the limit when A and B are at the same level), but always starts at a cusp. In contrast, the tautochrone problem can use only up to the first half rotation, and always ends at the horizontal.[2] The problem can be solved using tools from the calculus of variations[3] and optimal control.[4]
The curve is independent of both the mass of the test body and the local strength of gravity. Only a parameter is chosen so that the curve fits the starting point A and the ending point B.[5] If the body is given an initial velocity at A, or if friction is taken into account, then the curve that minimizes time differs from the tautochrone curve.
^Chisholm, Hugh, ed. (1911). "Brachistochrone" . Encyclopædia Britannica (11th ed.). Cambridge University Press.
^Stewart, James. "Section 10.1 - Curves Defined by Parametric Equations." Calculus: Early Transcendentals. 7th ed. Belmont, CA: Thomson Brooks/Cole, 2012. 640. Print.
^Weisstein, Eric W. "Brachistochrone Problem". MathWorld.
^Ross, I. M. The Brachistochrone Paradigm, in Primer on Pontryagin's Principle in Optimal Control, Collegiate Publishers, 2009. ISBN 978-0-9843571-0-9.
^Hand, Louis N., and Janet D. Finch. "Chapter 2: Variational Calculus and Its Application to Mechanics." Analytical Mechanics. Cambridge: Cambridge UP, 1998. 45, 70. Print.
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